Free Physics NEET Video Lectures for Waves

00:00:00 h [Music] having discussed oscillations in this lecture we want to start with waves so let us first understand what is a wave uh so question we raise is what is a w W is it particles moving from one place to the other so let's ask that question is it particles moving from one place to another let us explore that so you have seen waves like in an otion or if you take a big pan like a thali or something right and fill it with water and dip your finger in it so I'm dipping my finger here what you will see is that the water

00:01:50 goes up here comes down and this thing this Ripple uh let me give it the name Ripple travels down it travels does it take water along with it and we have seen it does not so only thing that travels is that Ripple and we call it a wave when I'm speaking to you or anybody across the room I create a disturbance here does it mean that the air goes from one place to the other if you stand in front of the person speaking you don't feel any air coming but you do hear him so what happens whatever disturbances

00:02:29 the person creates travels to your ear so when a person is speaking to another person sound travels but the air in between does not so certainly wave is not movement of a material from one place to the other so let us write that a wave does not represent movement of particles then what is it so the question again arises what is a wave and in simple terms what I'm going to say is wave a wave is a disturbance in any form I mean when I'm speaking to you I'm creating a disturbance in the air here and it

00:03:58 travels to the mic from there it gets recorded or if somebody is speaking to you that person is creating a disturbance in the air and that disturbance travels to you so wave is a disturbance that travels from one place to another but you have heard of standing waves right so question next one is does a wave mean that the disturbance has to travel from one place to another so this is another question and let me try to answer that suppose I have a string I tie it at one end and give a disturbance here give

00:05:02 it a jerk so what you will see is that this jerk or disturbance travels down the string so with time it travels this is certainly a traveling wave but think of this I tie this string between two ends and start shaking it you see that it gets distorted like like this and then comes down so it's moving up and down like this what we call a standing wave and in this wave the disturbance is not traveling so a wave could be a standing wave or a traveling or what we call a progressive wave or traveling wave so what we have answered

00:06:19 now is a wave is a disturbance created at one place and traveling to another place what I'll call the traveling waves or it could be and extended disturbance like the string and which I'll call the stationary wave and now I want to describe it quantitatively we want to see how really do we describe a wave so what we want to see is how do we describe a wave I'll be very general in this because what you have probably heard in your classes is a wave looks like this and it is traveling so what you

00:07:51 have seen is a disturbance yxt equals a sin 2 pi X over Lambda minus frequency ft this kind of wave or it can also be written in cosine terms a cine of 2 piun x/ Lambda minus ft and then you are told that it speed is v = f * Lambda where Lambda is the wavelength and F is the frequency this is a very particular kind of wave we'll come to it but before that since I called a wave is a traveling disturbance let me come to describe that first so let us say I have a string or as the example we gave is a

00:08:54 water surface and I create let's say a disturbance like this out here or I create a disturbance like this out here and it travels to understand it properly let me say that the disturbance I create is something like this at t equal to0 a parabolic one so let us say this is at time T = to 0 and the function f this is function f this is X Direction so function FX or since I'm going to later write y let me say y as a function of X looks like a² minus x² so what does it represent it represents at xal to 0

00:10:01 the height is a and it becomes zero at xal + a or minus a for mod x less than equal to a and it is zero otherwise just to give you a feeling for the pulse that we have created so this could also be a disturbance that has been created on the surface of water now this disturbance travels and let us say it has gone towards the right and has come at a point where the peak is at x0 then the same disturbance at a later time so let's say this has taken time t so at time T this disturbance would be

00:11:04 given as a s minus x - x² for mod x - x0 being less than equal to a and zero otherwise so what we have done is either on the surface of water on a string we created a disturbance and at a later time it moved to this point x knot and obviously if it travels with speed V this distance X KN is going to be VT and here it was given as YX = a² - x² for mod x less than equal to a zero otherwise and the right hand side I'm going to write this is in red y x at time T is given as a² - x - x0 s for mod x - x0 being less than equal to

00:12:12 a and 0 otherwise which can also be written as a s - x - VT squ for mod x minus VT less than equal to a and zero otherwise what about the case where it traveled to the left so suppose it was at xal to 0 and it traveled to the left if it travel to the left then yxt you can easily see is going to be a s minus x + VT s for x plus VT less than equal to a and zero otherwise because now this distance is going to be minus VT so what you see in both the cases that the disturbance y XT is a function of either x minus V t or x +

00:13:34 VT let us understand that so I'll make xal to 0 right here in the middle create some disturbance now I'm not even going to say it like a s - x but it's FX at T = to 0 then what would happen if it is traveling to the right in time T this disturbance is going to be given by f x - VT T = to Z why because now whatever disturbance is at time T which I'm going to call F XT is the same disturbance that was at x minus VT which was centered around xus VT at t equal to Z if on the other hand

00:14:30 it travel to the left then the peak to Peak distance would again be VT so whatever disturbance is at T as a function of X is the same as that which was at x + VT at t = 0 tal 0 is fixed so what we have shown is that any disturbance created at t equal to 0 around the origin it could be any shape if it travels with a constant speed V and undistorted so let me write this so a disturbance created as FX at time t equal to 0 will be change changing or will be given as a function of time as F XT equals f x minus

00:15:52 VT at tal to 0 if it travels to the right right means positive x-axis and it'll be given as F XT equals a function same function X Plus VT T = to 0 if it travels to the left because that's where it was now that's where it was at t equal to0 so a disturbance which has been created at t equal to 0 and disturbance I'm calling FX it could be any disturbance would appear as to the right I'm going to write f x - VT t = 0 equal fxt to the left I'm going to write FX + VT comma 0 equals fxt this has traveled

00:17:03 this way the green one has traveled to the right and under what conditions so conditions are the disturbance travels undistorted number one number two travels with constant speed V so two conditions it travels undistorted what is also called so undistorted also implies what is called in technical term dispersionless you'll hear this term quite often during the discussion on waves so it's a dispersionless travel it doesn't disperse it doesn't change shape it is undistorted and it travels with constant

00:18:13 speed V you can make out that undistorted assumption is really pretty okay when I speak to some person whether the person is standing at 10 m from me or 100 m from me or 200 M from me if he hear me or if she hears me what all these three persons standing at different distances here is the same word that I have spoken with the same interval and that means whatever disturbance I created goes pretty much undistorted so it's a pretty reasonable assumption there are cases in advanced courses you'll see where there is

00:18:51 dispersion one very obvious example of dispersion is light wave where the speed is different for different frequencies and that that's why you see a prism giving you different colors because the refractive index differs but right now we'll assume is dispersionless travel so that's one way of looking at it that is a function of xus VT or X Plus VT another way of looking at the wave travel another is that if I'm looking at a disturbance at certain point x then what I see exactly what happened at

00:19:46 t - x/ V at time okay another way of looking at the phenomena is way phenomena is let me suppose I'm standing at a point x and see a disturbance here this disturbance would have taken time T equals delta T equals x over V to travel from the origin this time taken is X over V so I can safely say that if I observe a disturbance at X at time T it would have been disturbance at xal 0 at a time t - x/ V this is another way of describing the wave phenomena if travel is to the right on the other hand if the

00:20:53 disturbance travels to the left then at XT what I see is something that would have occurred at X = to0 at time t + x over v x is less than zero so it's traveling to the left so I can also describe a wave as a function of t + x/ V or a function of t - x/ V and I'll just show by arrows - x over V is traveling to the right plus X over V is traveling to the left which is pretty much the same as writing it as xus VT and T minus X Plus VT because a function f of t + x over V is same as a function of VT + x over V which is a

00:21:52 function of x + VT remember the variables are x and t so V I can take as a constant and similarly function t - x / V is a function of VT - x / V which is same as a function of VT minus x v is a constant so I can write the wave disturbance traveling to the left to the right as X+ VT or a function of xus VT or a function of T minus X over V or a function of t + x/ V so let's write that so a wave can be described as a function of x + VT a function of x minus VT or a function of t + x over V or a function of T minus x/

00:22:56 V so all these forms exist and these are all traveling waves so let us see an example all you have seen is how do I generate yxt = a sin 2 pi x over Lambda minus ft this is what you have seen youve seen earlier a wave which is a disturbance like this and it travels I'll come to that in a minute but before that I again want to go back and discuss that these functions F of xus VT or FX + VT all these are functions of x and t both so a traveling or a standing wave is a function of x and t that is the position and time

00:24:05 both so if you want to visualize it you can either look at FX minus VT as a function of time by looking at a fixed position let's say x0 so what do I do I stand at one point and see as a function of Time how does the disturbance occur how does it change with time so I stand at X knot and see how wave disturbance is changing with time or what I can do is fix a t tal t 0 and observe how the disturbance looks as a function of X what would I do for that just take a photograph of the wave at that point and

00:25:32 then you can just see that as a function of X you will see that it is spread all right and it is at a particular time let's say t0 at a slightly later time it could be spread but somewhere else it would have traveled by distance VT or could have traveled to the left a distance VT undistorted so this is how I would look at a wave phenomena at a given time and at a given position if I stand there I'll just see that things are just moving up or down or left or right depending on what kind of wave it is and we will see how it

00:26:14 changes with time now let's see I said earlier that how is this wave generated right so a sin 2 pi X over Lambda minus ft so suppose I fix my position what do I see if I stand at a particular point then F at say x equal 0 as a function of time would look like a s of 2 piun x is 0/ Lambda minus fubt which which is a sin 2 piun ft with a minus sign so I can write this as minus a s of Omega T So as a function of time I'll just see that point at xal 0 going up and down what about as a function of

00:27:22 position as a function of position if I take a snapshot at a particular time this is what I'm going to see that this is spread from left to right all over and let's say the time equals 0 then it will look like FX at time tal to 0 is equal to a sin of 2 piun / Lambda X naturally what you see now is that if x goes to x + Lambda you get a sin 2 piun over Lambda x + Lambda which is again a sin 2 piun / Lambda x + 2 pi which is same as a sin 2i / Lambda X so after a distance Lambda the displacement is the same so the Lambda

00:28:22 which is the distance between two similar points because the displacement is the same I'm showing this is in pink this Lambda is called the wavelength so what I've shown you is that if I have this wave and I'll just take a portion of it this has wavelength Lambda between two similar points then the distance between two similar points is Lambda and after certain time T after certain time T it would have moved further so I'll just make it this distance would now be V delta T After

00:29:26 Time delta T now the next question I ask through this is how are Lambda V and frequency F of the wave related so let us try to answer that question I told you if I stand stand at a particular point x equal to 0 or x equals some other point x0 as a function of time this fellow is going to go up and down which goes as a sin of Omega T So after time T = capital T which is related to the frequency as 1 / f the displacement is the same now let's see if I took this wave Frozen in time and looked at

00:30:39 it after time capital T it look exactly the same except if there was a point a here this would have moved to point a at this point by distance Lambda so in time t the wave has moved by distance Lambda otherwise the displacement would not look the same so if it looks the same it must have moved by distance Lambda so V is going to be Lambda by T which is going to be f * Lambda this is is a relationship you very well know so the speed of wave is given by F * Lambda this particular wave which is known as the sinos

00:31:37 wave since we are talking about dispersionless waves the speed is the same for all frequencies how to generate a wave like a sin 2i x Lambda minus ft so let us look at that I'm now focusing specifically on this wave what is called is the sign wave so this is called uh S Wave so let's see if I have a string an extended string and I start shaking this point up and down as a s of Omega T so that time period is 2 pi/ Omega which is equal to T then as we had argued earlier so right now at Y at X = to 0 as

00:32:53 a function of time is equal to a sin of 2 pi ft then y as a function of X and time would be Y at x = 0 at time T minus X over V if it is traveling to the right and that I know is going to be a sin 2i f t - x over V which is equal to a sin 2i ft minus x f/ v and we have just seen the relationship that V equals F Lambda so I can write this as minus a sin of 2 pi X over Lambda minus fubt which is the same form that you have been seeing or which I showed you earlier so if I take a rope start shaking it at one end with with a sin

00:34:11 Omega T I create a S Wave which is traveling to the right if I did this disturbance and the wave traveled to the left then I would have y XT equals Y at x = 0 t + x/ V and this would be a sin 2 pi f t + x / V which is same as a sin 2i ft + x/ Lambda so these are different forms that you have seen so let me kind of summarize what we've done so far is that number one we have looked at a disturbance traveling with speed V and undistorted and therefore we could represent this as f x minus VT or F of T minus x/ V

00:35:40 if it travels to the right that means positive x-axis and it is represented as f x + VT or F of t + x/ V if traveling to the left so we've seen both ways all right and then we specialize to sinusoidal wave which is given as a y XT as amp itude a sin of x / Lambda - ft * 2 piun so it keeps repeating at on along the xaxis it keeps repeating at intervals of Lambda in time at a fixed point it the displacement keeps repeating itself in uh with frequency F in Time 1 / f and the way this is generated is by shaking at one

00:36:56 point with a given frequency in a simple harmonic manner so that the wave created something like this and is traveling either to the right or to the left we've also seen in this case that the speed V of the wave is given as the frequency time Lambda which can also be written as 2 pi frequency * Lambda over 2 pi this I can write as Omega omeg GA and now I'm introducing a new quantity 2 Pi / Lambda = K which is known as the wave vector or wave number 2 pi over Lambda is that many waves in 2 pi interval so Omega over K

00:37:49 so Omega equals V K that's a new relationship I'm giving you and in terms of this new relationship ship y XT can also be written as a sin of KX minus Omega T this is another form you will see in your books or places where waves are discussed so once we have understood this now let me give you two kinds of waves known as number one transverse waves these are waves where displacement y XT is perpendicular to the travel Direction so examples would be waves on a string are example of transvers

00:39:09 waves the other kind is longitudinal in which if the wave is traveling in the X Direction the disturbance is also along the same direction so in this the disturbance is in the same direction as the direction of motion of wave for example sound wave where I create a pressure difference that is the disturbance is an example of this so sound wave are long tudinal where the pressure is actually in the same direction as the propagation of wave one question we had asked in the beginning and I'm going to address it

00:40:20 now question do waves carry particles as they travel and the answer is no I had argued it earlier you can also see it and a string if you just shake it the string doesn't move from one place to the other the disturbance does if there's a water wave going you can leave a leaf or a piece of paper there and you'll see it's just moving up and down but it does not move with the wave so waves do do not carry particles as they move so they do not carry particles and also I can write do they carry

00:41:04 material and they do not question number two do waves carry energy from one place to another and the answer to this is yes the simplest possible way I can answer this is when I'm speaking to somebody the person hears whatever disturbance I have created whatever pressure difference I have created whatever movement local movement of particles I have created by speaking travels to the other place creates the same disturbance on the eard drum or in the ears that means it has the capability of taking

00:41:52 that energy from one place and transferring it to another place so yes waves do to carry energy from one place to other third question how do I calculate speed of waves and this is a question which I'm going to address now so let us first calculate the speed of waves on a string for this let's take a string and give it a disturbance let's say this is a disturbance a sign wave right and so let's say that y XT is given as a sin kx- Omega T and I know which I have said earlier that V the speed

00:43:12 is Omega over K or Omega equal VK I'm going to need this now let us see as a particular portion of this goes up and down what happens so if I take a particular piece of this string and assume so let's write on the side assume that Y is much much much less than Lambda this is tight so at a given time I've taken this snapshot I've taken a picture of this this FS ATT tension t to the right and it tension t to the left now this angle that it makes with the horizontal is very small so let's call this Theta here

00:44:11 and Theta here let's call this Theta 1 at Point 1 and Theta 2 at Point 2 so we going to say that Theta is much much much less than 1 implies sin Theta equal tan Theta equal Dy by DX for this given curve at this time and cos Theta is approximately equal to 1 so now this tension T at position two has two components one going up one going to the right this is T cos Theta which is roughly equal to T moving up is this t T sin Theta which is roughly equal to T Dy by DX at Point 2 on the lower side at point

00:45:12 one this is tension T it has a horizontal component and a vertical component the horizontal component again is given by T cosine of theta 1 which is roughly same as T and T sin Theta 1 in the vertical Direction which is T Dy by DX at 1 so on this section between 01 and point 2 I have net vertical force in the y direction which is T Dy by DX at this given time t d y by DX at 2 minus t Dy by DX at 1 in the y direction and zero in the X Direction let us say this distance between 1 and 2 is Delta X then by Taylor's theorem

00:46:28 or by simply taking derivatives I can write Dy by DX at 2 is equal to Dy by DX at 01 plus take the derivative of Dy by DX which will be D2 y by DX s Delta X therefore the vertical force on the section of this string is T d2y by DX s Delta x what put this Force do it'll make it accelerate up or down so if I equate this to the acceleration let's see what happens so acceleration of the section is going to be nothing but I'll freeze at that particular point at X and see it moving up and down so at that

00:47:39 given position I'll calculate Dy by DT ² that's the acceleration so the force which is T d2y by DX s at this given time which is also written as I'll just use it the technical term the partial derivative of y with respect to X because Y is a function of both x and t by writing partial I mean t is kept fixed and the acceleration is d2y by DT s at this position which is also written as partial derivative of y with respect to T this automatically means it's a given X and the relationship between the two

00:48:31 is going to be Force equals mass of the string Mu Delta X time acceleration where mu is mass per unit length of string so I get get T d2y by DX s at that fixed time is equal to Mu Delta X there's a Delta X here there was a Delta X on top D2 y by DT ² at fix X now let us take y XT equal a sin kx- us Omega T then what do we get so what we have written is that t d2y by DX s which correctly actually I should write as a partial derivative time Delta X is equal to Mu Delta X d2y by D t² at a given

00:49:49 X Delta X Delta X cancels and this is the equation and I'm taking a sinusoidal wave so y = a sin KX - Omega T d2y by DX Square for a fix time is going to be minus k s a s of KX - Omega T and d2y / DT ² a f x is going to be minus Omega s a sin of KX - Omega T and I substitute it back in this equation and I get t k s is equal to Mu Omega s and this gives me omega over K is equal to < TK of T / mu recall what we had said earlier earlier I had said that V equals Omega over K and this we have calculated to be

00:50:59 root t/ mu so you see how taking Newton's equation of motion for a particular section and relating how Omega and K should be related to the speed of wave that we did earlier although for a cidal wave we could get what the speed of wave should be in terms of T and mu so this also implies that wavelength for a given frequency will be V / f which will be 1 / f s Ro TK of T over mu so this is a derivation that we have done just by simply looking at how Newton equations are applied for a section of

00:51:44 wave another example I'm going to take calculation of speed of waves is the sound waves in this what we do is suppose I have a air column where I take a particular section let us say of length Delta X and create an extra pressure p at this point so when I'm speaking for example I create a pressure and this pressure changes with distance so it becomes P plus Delta p on the low next side in creating this pressure I have also moved this wall by a distance let's say Zed and therefore here on the other side

00:52:39 it moves by Zed plus Delta Zed so let me look at this shaded portion of air and look at this point which I'm showing by arrow and I'll see the movement of this hole shaded section as a hole and relate it C greatest acceleration related to the force that it is feeling now let us see the force it feels so if I look at this portion there's p + Delta p on this side p on this side so if this cross-sectional area is a then the force it feelds to the left which is Delta P times area and what would this Force too this

00:53:36 force would give it an acceleration so we are going to write that D2 Z by DT s at this point x let's say this point is x times the mass of of this portion shaded portion is going to be equal to minus Delta P time a from this I should be able to get the speed of the wave now what does this pressure also do so this Delta P the pressure difference makes it accelerate on the other hand the pressure p on both sides changes the volume through that volume change I can actually calculate relate Delta P to

00:54:41 some other property of this gas and let's do that so in this gas or in this air I have this small portion on which pressure p is here pressure p plus Delta p is here the length is Delta X I'm looking at Point Zed and what we have seen is that mass which is going to be the density of this air times this volume a Delta X then acceleration we have figured out is d2f by DT s then we've seen that the delta Delta P * a is the force and therefore we get row Delta X is equal to row Delta X time the acceleration d2f by

00:55:41 DT s is equal to minus Delta P * a there's an a also on the left hand side and this gets cancelled and you get row * d2f by DT ² is equal to minus Delta p over Delta X let us see how do we calculate this Delta p over Delta X so pressure p changes the volume mind you this pressure p is not the existing Ambiance pressure is the extra pressure that I'm creating so that's why this changes the volume how much does it ch change so I know that the bulk modulus is equal to minus V DP / DV or minus V whatever extra pressure

00:56:39 I'm applying and how much is the change in the I'm calling it Delta P bar over Delta V so Delta P bar is the pressure Delta P bar it's not the Delta P that I'm taking it's the pressure that I'm applying Delta V is going to be equal to initial volume is a Delta X the point on the left moves by Zed point on the Right Moves by Zed plus Delta Z so Delta V is going to be a zed plus Delta Zed minus a z which is nothing but a Delta Z which is nothing but a change in Zed as a function of x

00:57:41 times Delta X that's the change in the volume so therefore I'm going to have p over this change Delta V time V of this the sh ated area is equal to B is equal to minus V is a Delta X time pressure p divided by Delta V which is a d z by DX Delta X Delta X and Delta X cancels a and a cancels and I get therefore p is equal to minus b d d z over DX so what we have found is that row d2f by DT s is equal to minus Delta p/ Delta X which is like minus DP / DX and we have found that P is nothing

00:58:53 but equal to minus B dz/ DX what is z z the displacement of this what is f f is also displacement of the this left hand portion so f is same as Z which is giving me the acceleration of the body so I'm going to have row D2 Z / DT ² is equal to minus DP / DX X again this is no time so this is partial derivative which I can write now as plus b D2 z/ DX s and where did I get this from this I got from here so I have D2 Z / DX s is equal to row over B D2 z/ DT s or D2 z/ DT ² is equal to b/ row D2 z/ 01:

00:12 DX Square this is the equation we've got now what we saw earlier is that uh p is proportional to zed so I could also have written this in terms of P but this is what it is that is displacement that we create by changing P follows this equation and again taking sinos wave that means I'm taking Z XT to be a sin KX minus Omega T I get D2 Z by DX squ at a fixed time to be equal to minus a k 2 sin of KX - Omega T and D2 z/ DT ² at a fix X to be minus Omega s a s of KX - Omega T and substituting 01:01:16 these in the equation I get D2 F or D2 f/ DT s which is the same same as D2 z/ DT ² equals b/ row D2 z/ DX s or minus Omega s = b/ row time k s or Omega Square over k s which is nothing but v² equal B / row implies speed of waves is square root of B / R this is a well-known result now B is the bulk modulus and what has been argued is that at high frequencies this is the adiabetic bulk modulus because uh when there's a high frequency wave passing through there's not enough time for the heat to dissipate so it's 01:02:30 not constant temperature but tic bulk modulus so what we obtain now for that V for air or a material is square root of b/ row so let me just conclude this second portion by saying that we introduced the sinos waves then calculated speed of wave by relating acceleration of a portion of medium and we considered string and bulk medium for sound waves relating acceleration portion medium to to the force generated due to wave disturbance in the next coming lectures we will now explore these concepts related to waves further